Question

problem 7
the figure is a diagram of a wall. lengths are given in feet.
a. how many square feet of wallpaper would be needed to cover the wall? explain your reasoning.
b. wallpaper is sold in rolls that are 2 feet wide. what is the minimum length you would need to purchase to cover the wall?

Explanation:

Step1: Split the wall into shapes

The wall can be split into a rectangle and a triangle. The rectangle has length $48$ feet and height $9$ feet. The triangle has a base of $48$ feet and a height of $6$ feet.

Step2: Calculate the area of the rectangle

The area formula for a rectangle is $A = l\times w$. Here, $l = 48$ and $w=9$, so $A_{rectangle}=48\times9 = 432$ square - feet.

Step3: Calculate the area of the triangle

The area formula for a triangle is $A=\frac{1}{2}\times b\times h$. Here, $b = 48$ and $h = 6$, so $A_{triangle}=\frac{1}{2}\times48\times6=144$ square - feet.

Step4: Calculate the total area of the wall

The total area $A = A_{rectangle}+A_{triangle}=432 + 144=576$ square - feet.

Step5: Calculate the length of the wallpaper roll

The wallpaper roll is 2 feet wide. Let the length of the roll be $L$. The area of the wallpaper roll is $A_{roll}=L\times2$. We know that $A_{roll}$ must be at least equal to the area of the wall. So, $L\times2=576$, then $L=\frac{576}{2}=288$ feet.

Answer:

a. 576 square feet. The wall is composed of a rectangle with area $48\times9 = 432$ square - feet and a triangle with area $\frac{1}{2}\times48\times6 = 144$ square - feet. The total area is $432+144 = 576$ square - feet.
b. 288 feet. Since the wallpaper roll is 2 feet wide and the area of the wall is 576 square feet, the length of the roll needed is $\frac{576}{2}=288$ feet.